The extension and oscillation of a non-Hooke's law spring

We derive a wave equation for small-amplitude, undamped, extensional oscillation of a spring-mass system consisting of a mass suspended on a spring governed by a quadratic force-extension relationship. We justify this quadratic model using a Taylor series expansion of the general elasticity equations for a helical spring. Transformation of the equation of motion of the spring leads to a separable wave equation with the spacial component being a transformation of Bessel's equation. The model is successful in predicting static extension and period of oscillation of a helical wire spring for which the wave equation based on Hooke's law is inadequate.